Continuity and strict monotonicity of the exponent ω(β)

Determine whether the function ω(β) := lim_{x→∞} (log f_β(x) − x)/log x is continuous on [0,1) and whether it is strictly increasing, where f_β is the unique solution with f_β(0) = 1 to the functional equation ∫_0^∞ (f_β(sx)/f_β(x)) dx = 1/(1−s) − β for all s ∈ [0,1).

Background

The main result of the paper proves that the limit ω(β) exists for β ∈ [0,1), and an additional theorem shows that ω(β) is an increasing function with ω(β) ≤ 1. However, the authors indicate unresolved aspects concerning the regularity and strength of this monotonicity.

Clarifying continuity and strict monotonicity of ω(β) would sharpen the asymptotic characterization of the balanced model metric and further illuminate the dependence of the asymptotics on the parameter β.